2016 VCE Maths Methods Mini Test 2
Number of marks: 8
Reading time: 1 minute
Writing time: 12 minutes
Section B – Calculator Allowed
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In questions where a numerical answer is required, an exact value must be given unless otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Let \(f: [0, 8\pi] \to R, f(x) = 2\cos\left(\frac{x}{2}\right) + \pi\).
a. Find the period and range of \(f\). 2 marks
b. State the rule for the derivative function \(f'\). 1 mark
c. Find the equation of the tangent to the graph of \(f\) at \(x = \pi\). 1 mark
d. Find the equations of the tangents to the graph of \(f: [0, 8\pi] \to R, f(x) = 2\cos\left(\frac{x}{2}\right) + \pi\) that have a gradient of 1. 2 marks
e. The rule of \(f'\) can be obtained from the rule of \(f\) under a transformation \(T\), such that \(T: R^2 \to R^2, T\left(\begin{bmatrix} x \\ y \end{bmatrix}\right) = \begin{bmatrix} 1 & 0 \\ 0 & a \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} -\pi \\ b \end{bmatrix} \) Find the value of \(a\) and the value of \(b\). 3 marks
f. Find the values of \(x, 0 \le x \le 8\pi\), such that \(f(x) = 2f'(x) + \pi\). 2 marks
End of examination questions
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